Driskel accepted an athletic scholarship to attend the University of Florida, where he played for coach Will Muschamp's Florida Gators football team. As a freshman, Driskel appeared in five games, completing 16 of 34 passes for 148 yards with two interceptions. He became the team's starter as a sophomore in 2012. He led them to an 11–2 record and a berth in the 2013 Sugar Bowl, in which Florida lost to the Louisville Cardinals.[5]

After starting three games in 2013, Driskel broke his right fibula in a game against the University of Tennessee and missed the remainder of the season.[6] He returned from the injury in 2014 and kept his starting job to start the season but was later benched in favor of Treon Harris.[7][8]

Driskel was selected by the San Francisco 49ers in the sixth round of the 2016 NFL Draft with the 207th overall pick.[15] The 49ers previously traded tight end Vernon Davis to the Denver Broncos in exchange for the pick used to select Driskel. On May 5, 2016, the 49ers signed him to a four-year, $2.45 million contract with a signing bonus of $112,539.[16] Driskel began training camp as the No. 4 quarterback on the depth chart.[17] After a season-ending injury to Thad Lewis, Driskel competed with Christian Ponder for the third-string quarterback position. On September 3, 2016, he was waived by the 49ers.[18]

Driskel made his regular season NFL debut on October 21, 2018 in relief of Andy Dalton in the fourth quarter in the final minutes of a 45-10 loss to the Kansas City Chiefs, completing all four of his pass attempts for 39 yards.[21] In Week 12, against the Cleveland Browns, Dalton was injured and out for the season with a thumb injury.[22] Driskel came into the game in relief and threw his first professional touchdown to Tyler Boyd. In addition, he added a late rushing touchdown in the 35–20 loss.[23] He started the last five games of the season for the Bengals. He went 1–4 and passed for 764 yards, five touchdowns, and two interceptions in the five starts.[24]

Driskel was placed on injured reserve on August 31, 2019.[25] He was waived from injured reserve with an injury settlement on September 11.[26]